Lesson 17 - EngageNY
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Lesson 17: Sampling Variability Student Outcomes
Students use data from a random sample to estimate a population mean.
Students understand the term sampling variability in the context of estimating a population mean.
Lesson Notes Consider starting this lesson with a review of the activities from the last lesson. The total number of words for books in a random sample of children’s books was used to estimate the mean total number of words for the population of 150 children’s books. A random sample of prices from a population of grocery prices was used to estimate the proportion of grocery prices that would round up to the nearest dollar. This lesson and the lessons that follow continue to develop the idea of using data from a random sample to estimate a population mean or a population proportion. Sampling variability in sample means and sample proportions is explored in order to further student understanding of this important concept. In later grades, students learn how to take sampling variability into account when generalizing from a random sample to the population from which it was selected. In this lesson, students build on their developing understanding of sampling variability from the previous lessons by constructing a dot plot of the means of different random samples selected from the same population. In previous lessons, students have selected random samples by selecting items from a bag or by using a random number generator from a website or graphing calculator. This lesson adds another tool that can be used in selecting random samples—namely a random number table. A random number table is just a table of digits from 0 to 9 that are arranged in a random order. A random number table and directions for using it to select a random sample are provided as part of the lesson. Students may ask how the numbers in a random number table are generated. Today, computer programs are used to generate random number tables, but the idea behind these tables is relatively simple. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are placed in a bag or jar (with each digit written on a slip of paper or a chip). The 10 digits are thoroughly mixed. A slip or chip is picked. The number on the slip or chip is recorded and then returned to the bag or jar. This process is repeated, generating a random list of 0’s to 9’s.
Classwork Example 1 (5 minutes): Estimating a Population Mean Example 1: Estimating a Population Mean The owners of a gym have been keeping track of how long each person spends at the gym. Eight hundred of these times (in minutes) are shown in the population tables located at the end of the lesson. These 𝟖𝟖𝟖𝟖𝟖𝟖 times will form the population that you will investigate in this lesson.
Look at the values in the population. Can you find the longest time spent in the gym in the population? Can you find the shortest? On average, roughly how long do you think people spend at the gym? In other words, by just looking at the numbers in the two tables, make an estimate of the population mean.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
189 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
You could find the population mean by typing all 𝟖𝟖𝟖𝟖𝟖𝟖 numbers into a calculator or a computer, adding them up, and dividing by 𝟖𝟖𝟖𝟖𝟖𝟖. This would be extremely time-consuming, and usually it is not possible to measure every value in a population.
Instead of doing a calculation using every value in the population, we will use a random sample to find the mean of the sample. The sample mean will then be used as an estimate of the population mean.
In this example, students see a population of 800 times spent by people at a gym. The times are displayed in the population tables at the end of the lesson. Convey to students that it is not practical to use all 800 times to calculate the mean. It is important to make students aware of how unusual it would be to know all of the population values. Indeed, if you do have all the population values, then there is little point in taking a sample because you could calculate the population mean (using a computer if necessary). If there is time, ask the students to do the following:
Think of a population where you do not have all the population values.
Virtually any population will suffice in answer to this question; for example, the heights of all the adults in the town where the school is located.
Think of another population where you do have all the population values.
This is less common, but one example might be the salaries of all the employees of a company. All of the employee salaries will be stored in the company’s computer.
Discuss the questions in the example. Provide students time to think about and respond to the questions. The following responses provide a summary of the important points students should see as they work through the exercise.
Look at the values in the population. Can you find the longest time spent in the gym in the population? Can you find the shortest?
The longest time is 92 min., and the shortest time is 9 min. Students will provide other estimates, but direct them to these times.
On average, roughly how long do you think people spend at the gym? In other words, by just looking at the numbers in the two tables, make an estimate of the population mean.
It seems from the numbers in the tables that the mean time spent at the gym is around 60 min.
Example 2 (5 minutes): Selecting a Sample Using a Table of Random Digits Carefully discuss the sample directions outlined in this section. The goal is to recognize that each of the population values has an equal chance of being selected into the sample. Introduce the random number table, and as students work through this example, make sure that they understand how the table is used to identify individuals to be included in the sample. Once a sample is selected, the sample mean is calculated for this sample, and students are encouraged to think about how different samples would result in different values of the sample mean (this being an example of sampling variability).
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
190 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Example 2: Selecting a Sample Using a Table of Random Digits The table of random digits provided with this lesson will be used to select items from a population to produce a random sample from the population. The list of digits is determined by a computer program that simulates a random selection of the digits 𝟎𝟎, 𝟏𝟏, 𝟐𝟐, 𝟑𝟑, 𝟒𝟒, 𝟓𝟓, 𝟔𝟔, 𝟕𝟕, 𝟖𝟖, or 𝟗𝟗. Imagine that each of these digits is written on a slip of paper and placed in a bag. After thoroughly mixing the bag, one slip is drawn, and its digit is recorded in this list of random digits. The slip is then returned to the bag, and another slip is selected. The digit on this slip is recorded and then returned to the bag. The process is repeated over and over. The resulting list of digits is called a random number table.
Encourage students to think about the following question and to possibly share their suggestions prior to discussing the subsequent steps with all students. How could you use a table of random digits to take a random sample? Step 1: Place the table of random digits in front of you. Without looking at the page, place the eraser end of your pencil somewhere on the table. Start using the table of random digits at the number closest to where your eraser touched the paper. This digit and the following two specify which observation from the population tables will be the first observation in your sample. For example, suppose the eraser end of your pencil lands on the twelfth number in row 𝟑𝟑 of the random digit table. This number is 𝟓𝟓, and the two following numbers are 𝟏𝟏 and 𝟒𝟒. This means that the first observation in your sample is observation number 𝟓𝟓𝟓𝟓𝟓𝟓 from the population. Find observation number 𝟓𝟓𝟓𝟓𝟓𝟓 in the population table. Do this by going to Row 𝟓𝟓𝟓𝟓 and moving across to the column heading “𝟒𝟒.” This observation is 𝟓𝟓𝟓𝟓, so the first observation in your sample is 𝟓𝟓𝟓𝟓. If the number from the random number table is any number 𝟖𝟖𝟖𝟖𝟖𝟖 or greater, you will ignore this number and use the next three digits in the table.
Point out that four more selections are to be made to complete the random sample of five. Emphasize that the next four observations are determined by continuing on from the starting position in the random number table. The following directions and observations are important for students to understand: Step 2: Continue using the table of random digits from the point you reached, and select the other four observations in your sample like you did above. For example, continuing on from the position in the example given in Step 1:
The next number from the random digit table is 𝟕𝟕𝟕𝟕𝟕𝟕, and observation 𝟕𝟕𝟕𝟕𝟕𝟕 is 𝟔𝟔𝟔𝟔
The next number from the random digit table is 𝟓𝟓𝟓𝟓𝟓𝟓, and observation 𝟓𝟓𝟓𝟓𝟓𝟓 is 𝟑𝟑𝟑𝟑.
The next number from the random digit table is 𝟗𝟗𝟗𝟗𝟗𝟗, and there is no observation 𝟗𝟗𝟗𝟗𝟗𝟗. So, we ignore these three digits.
The next number from the random digit table is 𝟗𝟗𝟗𝟗𝟗𝟗, and there is no observation 𝟗𝟗𝟗𝟗𝟗𝟗. So, we ignore these three digits. The next number from the random digit table is 𝟑𝟑𝟑𝟑𝟑𝟑, and observation 𝟑𝟑𝟑𝟑𝟑𝟑 is 𝟕𝟕𝟕𝟕.
The next number from the random digit table is 𝟔𝟔𝟔𝟔𝟔𝟔, and observation 𝟔𝟔𝟔𝟔𝟔𝟔 is 𝟒𝟒𝟒𝟒.
Exercises 1–4 (8 minutes) Students should work independently on Exercises 1–4. Then, discuss and confirm as a class. The values found in Example 2 are used to illustrate the answers to the following questions, but students should select their own random sample of five.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
191 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Exercises 1–4 Initially, you will select just five values from the population to form your sample. This is a very small sample size, but it is a good place to start to understand the ideas of this lesson. 1.
Use the table of random numbers to select five values from the population of times. What are the five observations in your sample? Make sure students write down these values and understand how they were selected. In the example given, the following observations were selected: 𝟓𝟓𝟓𝟓 𝐦𝐦𝐦𝐦𝐦𝐦., 𝟔𝟔𝟔𝟔 𝐦𝐦𝐦𝐦𝐦𝐦., 𝟑𝟑𝟑𝟑 𝐦𝐦𝐦𝐦𝐦𝐦., 𝟕𝟕𝟕𝟕 𝐦𝐦𝐦𝐦𝐦𝐦., and 𝟒𝟒𝟒𝟒 𝐦𝐦𝐦𝐦𝐦𝐦.
2.
For the sample that you selected, calculate the sample mean. For the given example, the sample mean is
3.
𝟓𝟓𝟓𝟓+𝟔𝟔𝟔𝟔+𝟑𝟑𝟑𝟑+𝟕𝟕𝟕𝟕+𝟒𝟒𝟒𝟒 𝟓𝟓
= 𝟓𝟓𝟓𝟓. 𝟖𝟖 𝐦𝐦𝐦𝐦𝐦𝐦.
You selected a random sample and calculated the sample mean in order to estimate the population mean. Do you think that the mean of these five observations is exactly correct for the population mean? Could the population mean be greater than the number you calculated? Could the population mean be less than the number you calculated? Make sure that students see that the value of their sample mean (𝟓𝟓𝟓𝟓. 𝟖𝟖 minutes in the given example, but students will have different sample means) is not likely to be exactly correct for the population mean. The population mean could be greater or less than the value of the sample mean.
4.
In practice, you only take one sample in order to estimate a population characteristic. But, for the purposes of this lesson, suppose you were to take another random sample from the same population of times at the gym. Could the new sample mean be closer to the population mean than the mean of these five observations? Could it be farther from the population mean? Make sure students understand that if they were to take a new random sample, the new sample mean is unlikely to be equal to the value of the sample mean of their first sample. It could be closer to or further from the population mean.
Exercises 5–7 (7 minutes) Before students begin the following exercises, put all of their sample means on the board or poster paper. Then, ask the following questions: What do you notice about the sample means? Why do you think this is true? Anticipate that students will indicate that the sample means posted are different and that several of them are very close to each other. They may also point out how different some of them are from each other. Students should recognize that the samples come from the same population but that the actual selections can often be different. The different selections result in different sample means. Point out that the sample mean is an example of a sample statistic. The variation of sample statistics from sample to sample is called sampling variability.
Here, each student selects a random sample from the population and calculates the sample mean. Students should work individually, helping their neighbors as needed. The use of a calculator to calculate the sample mean should be encouraged to save time and increase accuracy. Point out that for this exercise, students should determine the mean to the nearest tenth of a minute.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
192 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Exercises 5–7 As a class, you will now investigate sampling variability by taking several samples from the same population. Each sample will have a different sample mean. This variation provides an example of sampling variability. 5.
Place the table of random digits in front of you, and without looking at the page, place the eraser end of your pencil somewhere on the table of random numbers. Start using the table of random digits at the number closest to where your eraser touches the paper. This digit and the following two specify which observation from the population tables will be the first observation in your sample. Write this three-digit number and the corresponding data value from the population in the space below. Answers will vary based on the numbers selected.
6.
Continue moving to the right in the table of random digits from the place you ended in Exercise 5. Use three digits at a time. Each set of three digits specifies which observation in the population is the next number in your sample. Continue until you have four more observations, and write these four values in the space below. Answers will vary based on the numbers selected.
7.
Calculate the mean of the five values that form your sample. Round your answer to the nearest tenth. Show your work and your sample mean in the space below. Answers will vary based on the numbers selected.
Exercises 8–11 (10 minutes) Exercises 8–11 You will now use the sample means from Exercise 7 from the entire class to make a dot plot. 8.
Write the sample means for everyone in the class in the space below. Answers will vary based on collected sample means.
9.
Use all the sample means to make a dot plot using the axis given below. (Remember, if you have repeated or close values, stack the dots one above the other.)
35
40
45
50 55 Sample Mean
60
65
70
10. What do you see in the dot plot that demonstrates sampling variability? Sample response: The dots are spread out indicating that the sample means are not the same. The results indicate what we discussed as sampling variability. (See explanation provided at the end of this lesson.)
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
193 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 17
7•5
11. Remember that in practice you only take one sample. (In this lesson, many samples were taken in order to demonstrate the concept of sampling variability.) Suppose that a statistician plans to take a random sample of size 𝟓𝟓 from the population of times spent at the gym and that he will use the sample mean as an estimate of the population mean. Approximately how far can the statistician expect the sample mean to be from the population mean? Answers will vary. Allow students to speculate as to what the value of the population mean might be and how far a sample mean would be from that value. The distance students indicate will be based on the dot plot. Students may indicate the sample mean to be exactly equal to the population and that the distance would be 𝟎𝟎. They may also indicate the sample mean to be one of the dots that is a minimum or a maximum of the distribution and suggest a distance from the minimum or maximum to the center of the distribution. Students should begin to see that the distribution has a center that they suspect is close to the population’s mean.
Here, the sample means for the class are combined in order to form a visual illustration of sampling variability (in the MP.2 form of a dot plot of the class’s sample means). Students are asked to show how the pattern in the dot plot supports the notion of sampling variability. Students will need to refer to the dot plot in their work in the next lesson, so every student will need to complete the dot plot. Exercise 10 summarizes an important part of this lesson. The fact that the sample mean varies from sample to sample is an example of sampling variability. Clearly emphasize this summary. Exercise 11 covers an important concept that is worth thinking about in advance of the class. When you take a sample and calculate the sample mean, you know that the value of the sample mean is very unlikely to be exactly equal to the population mean. But how close can you expect it to be? The dot plot constructed in this lesson provides a way to answer this question (at least for a random sample of size 5 from the population presented in this class). The population mean is approximately in the center of the dots in the dot plot constructed in Exercise 9. Indicate that the distances of the dots from the population mean provide information about how far the statistician can expect the sample mean to be from the population mean. An important point is also made at the beginning of Exercise 11, which is worth emphasizing whenever the opportunity arises in the remaining lessons. In practice, you only take one sample from a population. If you took two samples, then the sensible thing would be to combine them into one larger sample. We only take multiple samples in this lesson in order to demonstrate the concept of sampling variability.
Suppose that you are going to take one sample from a population. Once you have your sample, you will calculate the sample mean. Explain what is meant by sampling variability in this context.
Before taking your sample, you know that there are many different sets of individuals who could form your sample. Every different possible sample could give a different value for the sample mean. This is known as sampling variability.
It is important that students be aware that the idea of sampling variability does not apply only to the sample mean. Suppose, for example, an advertising agency plans to take a random sample of people and ask those people whether they have seen a particular commercial. The population is all the people who could be asked the question. The proportion of people in the sample who have seen the commercial is a sample statistic, and the value of this sample statistic will vary from sample to sample. This statistic also has sampling variability. Simulations will be developed in later lessons to illustrate sampling variability of proportions calculated using data from random samples of a population. Lesson 20 begins students’ investigations with the proportion as the statistic of interest.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
194 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Population
𝟏𝟏
𝟐𝟐
𝟑𝟑
𝟒𝟒
𝟓𝟓
𝟔𝟔
𝟕𝟕
𝟖𝟖
𝟗𝟗
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟎𝟎𝟎𝟎
𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒
𝟖𝟖𝟖𝟖
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟖𝟖𝟖𝟖
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 Lesson 17: Date:
𝟎𝟎
𝟎𝟎𝟎𝟎
𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐
𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐
𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖
𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔 𝟐𝟐𝟐𝟐 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟗𝟗𝟗𝟗 𝟗𝟗
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟗𝟗𝟗𝟗 𝟗𝟗
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
195 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Population, continued
𝟎𝟎
𝟐𝟐
𝟑𝟑
𝟒𝟒
𝟓𝟓
𝟔𝟔
𝟕𝟕
𝟖𝟖
𝟗𝟗
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑
𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓
𝟏𝟏𝟏𝟏
𝟓𝟓𝟓𝟓
𝟖𝟖𝟖𝟖
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟔𝟔𝟔𝟔
𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏
𝟔𝟔𝟔𝟔
𝟖𝟖𝟖𝟖
𝟕𝟕𝟕𝟕
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟕𝟕𝟕𝟕
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟔𝟔𝟔𝟔
𝟕𝟕𝟕𝟕
Lesson 17: Date:
𝟏𝟏
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑
𝟖𝟖𝟖𝟖 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟏𝟏𝟏𝟏 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟏𝟏𝟏𝟏 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟏𝟏𝟏𝟏 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟏𝟏𝟏𝟏 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟏𝟏𝟏𝟏
𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔
𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
196 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Table of Random Digits Row 𝟏𝟏
𝟔𝟔
𝟔𝟔
𝟕𝟕
𝟐𝟐
𝟖𝟖
𝟎𝟎
𝟎𝟎
𝟖𝟖
𝟒𝟒
𝟎𝟎
𝟎𝟎
𝟒𝟒
𝟔𝟔
𝟎𝟎
𝟑𝟑
𝟐𝟐
𝟐𝟐
𝟒𝟒
𝟔𝟔
𝟖𝟖
𝟒𝟒
𝟗𝟗
𝟏𝟏
𝟏𝟏
𝟗𝟗
𝟐𝟐
𝟖𝟖
𝟑𝟑
𝟎𝟎
𝟑𝟑
𝟔𝟔
𝟕𝟕
𝟕𝟕
𝟒𝟒
𝟕𝟕
𝟓𝟓
𝟗𝟗
𝟖𝟖
𝟏𝟏
𝟖𝟖
𝟑𝟑
𝟐𝟐 𝟑𝟑 𝟓𝟓 𝟔𝟔 𝟕𝟕 𝟖𝟖 𝟗𝟗
𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒
𝟖𝟖 𝟓𝟓 𝟗𝟗 𝟖𝟖 𝟒𝟒 𝟗𝟗 𝟑𝟑 𝟐𝟐 𝟎𝟎
𝟖𝟖 𝟒𝟒 𝟒𝟒 𝟔𝟔 𝟑𝟑 𝟔𝟔 𝟖𝟖 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟖𝟖 𝟒𝟒 𝟗𝟗 𝟑𝟑 𝟐𝟐 𝟎𝟎
𝟖𝟖 𝟒𝟒 𝟒𝟒 𝟔𝟔 𝟑𝟑 𝟔𝟔 𝟖𝟖 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟖𝟖 𝟒𝟒 𝟗𝟗
𝟎𝟎
𝟑𝟑 𝟎𝟎
𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟐𝟐 𝟔𝟔 𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟐𝟐 𝟔𝟔 𝟎𝟎
𝟑𝟑 𝟏𝟏 𝟎𝟎
𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟐𝟐 𝟔𝟔 𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟐𝟐 𝟔𝟔 𝟎𝟎
𝟑𝟑 𝟏𝟏 𝟎𝟎
𝟏𝟏 𝟏𝟏 𝟗𝟗
Lesson 17: Date:
𝟑𝟑 𝟓𝟓 𝟐𝟐 𝟒𝟒 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟓𝟓 𝟎𝟎
𝟔𝟔 𝟗𝟗 𝟖𝟖 𝟑𝟑 𝟕𝟕 𝟑𝟑 𝟓𝟓 𝟏𝟏 𝟐𝟐 𝟒𝟒 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟓𝟓 𝟎𝟎
𝟔𝟔 𝟗𝟗 𝟖𝟖 𝟑𝟑 𝟕𝟕 𝟑𝟑 𝟓𝟓 𝟏𝟏 𝟐𝟐 𝟒𝟒 𝟏𝟏 𝟐𝟐
𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟔𝟔 𝟗𝟗 𝟕𝟕 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟒𝟒 𝟖𝟖 𝟎𝟎
𝟖𝟖 𝟐𝟐 𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟗𝟗 𝟔𝟔 𝟗𝟗 𝟕𝟕 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟒𝟒 𝟖𝟖 𝟎𝟎
𝟖𝟖 𝟐𝟐 𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟗𝟗 𝟔𝟔 𝟗𝟗 𝟕𝟕
𝟏𝟏 𝟑𝟑 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟏𝟏 𝟗𝟗 𝟓𝟓 𝟎𝟎
𝟒𝟒 𝟓𝟓 𝟕𝟕 𝟎𝟎
𝟒𝟒 𝟖𝟖 𝟏𝟏 𝟑𝟑 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟏𝟏 𝟗𝟗 𝟓𝟓 𝟎𝟎
𝟒𝟒 𝟓𝟓 𝟕𝟕 𝟎𝟎
𝟒𝟒 𝟖𝟖 𝟏𝟏 𝟑𝟑 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟏𝟏
𝟏𝟏 𝟔𝟔 𝟕𝟕 𝟔𝟔 𝟎𝟎
𝟑𝟑 𝟏𝟏 𝟗𝟗 𝟖𝟖 𝟖𝟖 𝟐𝟐 𝟑𝟑 𝟎𝟎
𝟖𝟖 𝟎𝟎
𝟏𝟏 𝟔𝟔 𝟖𝟖 𝟕𝟕 𝟔𝟔 𝟎𝟎
𝟑𝟑 𝟏𝟏 𝟗𝟗 𝟖𝟖 𝟖𝟖 𝟐𝟐 𝟑𝟑 𝟎𝟎
𝟖𝟖 𝟎𝟎
𝟏𝟏 𝟔𝟔 𝟖𝟖 𝟕𝟕 𝟔𝟔 𝟎𝟎
𝟑𝟑
𝟏𝟏 𝟑𝟑 𝟒𝟒 𝟖𝟖 𝟕𝟕 𝟐𝟐 𝟗𝟗 𝟗𝟗 𝟏𝟏 𝟖𝟖 𝟒𝟒 𝟒𝟒 𝟒𝟒 𝟖𝟖 𝟎𝟎
𝟏𝟏 𝟑𝟑 𝟑𝟑 𝟒𝟒 𝟖𝟖 𝟕𝟕 𝟐𝟐 𝟗𝟗 𝟗𝟗 𝟏𝟏 𝟖𝟖 𝟒𝟒 𝟒𝟒 𝟒𝟒 𝟖𝟖 𝟎𝟎
𝟏𝟏 𝟑𝟑 𝟑𝟑 𝟒𝟒 𝟖𝟖 𝟕𝟕 𝟐𝟐
𝟐𝟐 𝟏𝟏 𝟔𝟔 𝟐𝟐 𝟐𝟐 𝟗𝟗 𝟎𝟎 𝟑𝟑 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟐𝟐 𝟔𝟔 𝟖𝟖 𝟐𝟐 𝟏𝟏 𝟎𝟎 𝟔𝟔 𝟐𝟐 𝟐𝟐 𝟗𝟗 𝟎𝟎 𝟑𝟑 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟐𝟐 𝟔𝟔 𝟖𝟖 𝟐𝟐 𝟏𝟏 𝟎𝟎 𝟔𝟔 𝟐𝟐 𝟐𝟐 𝟗𝟗
𝟕𝟕 𝟕𝟕 𝟑𝟑 𝟖𝟖 𝟗𝟗
𝟎𝟎 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕 𝟔𝟔 𝟖𝟖 𝟑𝟑 𝟐𝟐 𝟒𝟒 𝟕𝟕 𝟕𝟕 𝟑𝟑 𝟑𝟑 𝟖𝟖 𝟗𝟗
𝟎𝟎 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕 𝟔𝟔 𝟖𝟖 𝟑𝟑 𝟐𝟐 𝟒𝟒 𝟕𝟕 𝟕𝟕 𝟑𝟑 𝟑𝟑 𝟖𝟖 𝟗𝟗
𝟎𝟎
𝟎𝟎 𝟐𝟐 𝟔𝟔 𝟗𝟗 𝟎𝟎 𝟑𝟑 𝟖𝟖 𝟗𝟗 𝟕𝟕 𝟕𝟕 𝟏𝟏 𝟐𝟐 𝟖𝟖 𝟗𝟗 𝟎𝟎 𝟎𝟎 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟗𝟗 𝟎𝟎 𝟑𝟑 𝟖𝟖 𝟗𝟗 𝟕𝟕 𝟕𝟕 𝟏𝟏 𝟐𝟐 𝟖𝟖 𝟗𝟗 𝟎𝟎 𝟎𝟎 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟗𝟗 𝟎𝟎 𝟑𝟑
𝟏𝟏 𝟓𝟓 𝟔𝟔 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟏𝟏 𝟑𝟑 𝟏𝟏
𝟎𝟎 𝟕𝟕 𝟗𝟗 𝟏𝟏 𝟏𝟏
𝟎𝟎 𝟏𝟏 𝟓𝟓 𝟕𝟕 𝟔𝟔 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟏𝟏 𝟑𝟑 𝟏𝟏
𝟎𝟎 𝟕𝟕 𝟗𝟗 𝟏𝟏 𝟏𝟏
𝟎𝟎 𝟏𝟏 𝟓𝟓 𝟕𝟕 𝟔𝟔 𝟓𝟓 𝟗𝟗 𝟗𝟗
𝟗𝟗 𝟓𝟓 𝟑𝟑 𝟓𝟓 𝟕𝟕 𝟎𝟎 𝟔𝟔 𝟗𝟗 𝟑𝟑 𝟏𝟏 𝟓𝟓 𝟏𝟏 𝟖𝟖 𝟎𝟎 𝟒𝟒 𝟗𝟗 𝟓𝟓 𝟕𝟕 𝟑𝟑 𝟓𝟓 𝟕𝟕 𝟎𝟎 𝟔𝟔 𝟗𝟗 𝟑𝟑 𝟏𝟏 𝟓𝟓 𝟏𝟏 𝟖𝟖 𝟎𝟎 𝟒𝟒 𝟗𝟗 𝟓𝟓 𝟕𝟕 𝟑𝟑 𝟓𝟓 𝟕𝟕 𝟎𝟎
𝟏𝟏 𝟏𝟏 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕 𝟐𝟐 𝟏𝟏
𝟎𝟎 𝟎𝟎 𝟔𝟔 𝟓𝟓 𝟒𝟒 𝟏𝟏 𝟔𝟔 𝟏𝟏 𝟏𝟏 𝟒𝟒 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕 𝟐𝟐 𝟏𝟏
𝟎𝟎 𝟎𝟎 𝟔𝟔 𝟓𝟓 𝟒𝟒 𝟏𝟏 𝟔𝟔 𝟏𝟏 𝟏𝟏 𝟒𝟒 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕
𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟓𝟓 𝟗𝟗 𝟒𝟒 𝟏𝟏 𝟔𝟔 𝟑𝟑 𝟎𝟎 𝟗𝟗 𝟎𝟎 𝟐𝟐 𝟒𝟒 𝟕𝟕 𝟒𝟒 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟓𝟓 𝟗𝟗 𝟒𝟒 𝟏𝟏 𝟔𝟔 𝟑𝟑 𝟎𝟎 𝟗𝟗 𝟎𝟎 𝟐𝟐 𝟒𝟒 𝟕𝟕 𝟒𝟒 𝟗𝟗 𝟒𝟒 𝟓𝟓
𝟕𝟕 𝟕𝟕 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟔𝟔 𝟑𝟑
𝟎𝟎 𝟑𝟑 𝟕𝟕 𝟒𝟒 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟑𝟑 𝟕𝟕 𝟕𝟕 𝟓𝟓 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟔𝟔 𝟑𝟑
𝟎𝟎 𝟑𝟑 𝟕𝟕 𝟒𝟒 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟑𝟑 𝟕𝟕 𝟕𝟕 𝟓𝟓 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟔𝟔
𝟏𝟏 𝟏𝟏 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟕𝟕 𝟗𝟗 𝟓𝟓 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟗𝟗 𝟓𝟓 𝟑𝟑 𝟐𝟐 𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟕𝟕 𝟗𝟗 𝟓𝟓 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟗𝟗 𝟓𝟓 𝟑𝟑 𝟐𝟐 𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟕𝟕
𝟑𝟑 𝟔𝟔 𝟎𝟎 𝟓𝟓 𝟑𝟑 𝟏𝟏
𝟎𝟎 𝟓𝟓 𝟔𝟔 𝟑𝟑 𝟖𝟖 𝟖𝟖
𝟎𝟎 𝟎𝟎 𝟐𝟐 𝟑𝟑 𝟔𝟔 𝟖𝟖
𝟎𝟎 𝟓𝟓 𝟑𝟑 𝟏𝟏
𝟎𝟎 𝟓𝟓 𝟔𝟔 𝟑𝟑 𝟖𝟖 𝟖𝟖
𝟎𝟎 𝟎𝟎 𝟐𝟐 𝟑𝟑 𝟔𝟔 𝟖𝟖
𝟎𝟎 𝟓𝟓 𝟑𝟑 𝟏𝟏
𝟑𝟑 𝟓𝟓 𝟎𝟎 𝟑𝟑 𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟏𝟏 𝟒𝟒 𝟓𝟓 𝟒𝟒 𝟗𝟗 𝟗𝟗 𝟕𝟕 𝟒𝟒 𝟑𝟑 𝟓𝟓 𝟏𝟏 𝟎𝟎 𝟑𝟑 𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟏𝟏 𝟒𝟒 𝟓𝟓 𝟒𝟒 𝟗𝟗 𝟗𝟗 𝟕𝟕 𝟒𝟒 𝟑𝟑 𝟓𝟓 𝟏𝟏 𝟎𝟎 𝟑𝟑 𝟏𝟏 𝟕𝟕
𝟓𝟓 𝟔𝟔 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟖𝟖 𝟏𝟏 𝟒𝟒 𝟒𝟒 𝟑𝟑 𝟏𝟏 𝟐𝟐 𝟎𝟎 𝟑𝟑 𝟔𝟔 𝟓𝟓 𝟔𝟔 𝟖𝟖 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟖𝟖 𝟏𝟏 𝟒𝟒 𝟒𝟒 𝟑𝟑 𝟏𝟏 𝟐𝟐 𝟎𝟎 𝟑𝟑 𝟔𝟔 𝟓𝟓 𝟔𝟔 𝟖𝟖 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟖𝟖
𝟑𝟑 𝟓𝟓 𝟗𝟗 𝟑𝟑 𝟗𝟗 𝟕𝟕
𝟎𝟎 𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟖𝟖 𝟑𝟑 𝟓𝟓 𝟑𝟑 𝟗𝟗 𝟑𝟑 𝟗𝟗 𝟕𝟕
𝟎𝟎 𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟖𝟖 𝟑𝟑 𝟓𝟓 𝟑𝟑 𝟗𝟗 𝟑𝟑 𝟗𝟗 𝟕𝟕
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
197 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 17
7•5
Closing (4 minutes) If there is time, ask students these questions:
You have learned today about sampling variability. Do you want sampling variability to be large or small?
We want sampling variability to be small.
Why do you want sampling variability to be small?
Suppose you are using the sample mean to estimate a population mean. You want the sampling variability of the sample mean to be small, so that you can expect your value of the sample mean to be close to the population mean.
Lesson Summary A population characteristic is estimated by taking a random sample from the population and calculating the value of a statistic for the sample. For example, a population mean is estimated by selecting a random sample from the population and calculating the sample mean. The value of the sample statistic (e.g., the sample mean) will vary based on the random sample that is selected. This variation from sample to sample in the values of the sample statistic is called sampling variability.
Exit Ticket (6 minutes)
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
198 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ___________________________________________________
7•5
Date____________________
Lesson 17: Sampling Variability Exit Ticket Suppose that you want to estimate the mean time per evening students at your school spend doing homework. You will do this using a random sample of 30 students.
1.
Suppose that you have a list of all the students at your school. The students are numbered 1, 2, 3, …. One way to select the random sample of students is to use the random digit table from today’s class, taking three digits at a time. If you start at the third digit of row 9, what is the number of the first student you would include in your sample?
2.
Suppose that you have now selected your random sample and that you have asked the students how long they spend doing homework each evening. How will you use these results to estimate the mean time spent doing homework for all students?
3.
Explain what is meant by sampling variability in this context.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
199 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Exit Ticket Sample Solutions Suppose that you want to estimate the mean time per evening students at your school spend doing homework. You will do this using a random sample of 𝟑𝟑𝟑𝟑 students.
1.
Suppose that you have a list of all the students at your school. The students are numbered 𝟏𝟏, 𝟐𝟐, 𝟑𝟑, …. One way to select the random sample of students is to use the random digit table from today’s class, taking three digits at a time. If you start at the third digit of row 𝟗𝟗, what is the number of the first student you would include in your sample? The first student in the sample would be student number 𝟐𝟐𝟐𝟐𝟐𝟐.
2.
Suppose that you have now selected your random sample and that you have asked the students how long they spend doing homework each evening. How will you use these results to estimate the mean time spent doing homework for all students? I would calculate the mean time spent doing homework for the students in my sample.
3.
Explain what is meant by sampling variability in this context. Different samples of students would result in different values of the sample mean. This is sampling variability of the sample mean.
Problem Set Sample Solutions 1.
Yousef intends to buy a car. He wishes to estimate the mean fuel efficiency (in miles per gallon) of all cars available at this time. Yousef selects a random sample of 𝟏𝟏𝟏𝟏 cars and looks up their fuel efficiencies on the Internet. The results are shown below.
a.
b.
𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐
Yousef will estimate the mean fuel efficiency of all cars by calculating the mean for his sample. Calculate the sample mean, and record your answer below. (Be sure to show your work.) 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟑𝟑𝟑𝟑 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 = 𝟐𝟐𝟐𝟐. 𝟗𝟗 𝟏𝟏𝟏𝟏
In practice, you only take one sample to estimate a population characteristic. However, if Yousef were to take another random sample of 𝟏𝟏𝟏𝟏 cars from the same population, would he likely get the same value for the sample mean? No, it is not likely that Yousef would get the same value for the sample mean.
c.
What if Yousef were to take many random samples of 𝟏𝟏𝟏𝟏 cars? Would all of the sample means be the same? No, he could get many different values of the sample mean.
d.
Using this example, explain what sampling variability means. The fact that the sample mean will vary from sample to sample is an example of sampling variability.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
200 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
7•5
Think about the mean number of siblings (brothers and sisters) for all students at your school. a.
What do you think is the approximate value of the mean number of siblings for the population of all students at your school? Answers will vary.
b.
How could you find a better estimate of this population mean? I could take a random sample of students, ask the students in my sample how many siblings they have, and find the mean for my sample.
c.
Suppose that you have now selected a random sample of students from your school. You have asked all of the students in your sample how many siblings they have. How will you calculate the sample mean? I will add up all of the values in the sample and divide by the number of students in the sample.
d.
If you had taken a different sample, would the sample mean have taken the same value? No. A different sample would generally produce a different value of the sample mean. It is possible, but unlikely, that the sample mean for a different sample would have the same mean.
e.
There are many different samples of students that you could have selected. These samples produce many different possible sample means. What is the phrase used for this concept? Sampling variability.
f.
Does the phrase you gave in part (e) apply only to sample means? No. The concept of sampling variability applies to any sample statistic.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
201 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Lesson 17: Sampling Variability Student Outcomes
Students use data from a random sample to estimate a population mean.
Students understand the term sampling variability in the context of estimating a population mean.
Lesson Notes Consider starting this lesson with a review of the activities from the last lesson. The total number of words for books in a random sample of children’s books was used to estimate the mean total number of words for the population of 150 children’s books. A random sample of prices from a population of grocery prices was used to estimate the proportion of grocery prices that would round up to the nearest dollar. This lesson and the lessons that follow continue to develop the idea of using data from a random sample to estimate a population mean or a population proportion. Sampling variability in sample means and sample proportions is explored in order to further student understanding of this important concept. In later grades, students learn how to take sampling variability into account when generalizing from a random sample to the population from which it was selected. In this lesson, students build on their developing understanding of sampling variability from the previous lessons by constructing a dot plot of the means of different random samples selected from the same population. In previous lessons, students have selected random samples by selecting items from a bag or by using a random number generator from a website or graphing calculator. This lesson adds another tool that can be used in selecting random samples—namely a random number table. A random number table is just a table of digits from 0 to 9 that are arranged in a random order. A random number table and directions for using it to select a random sample are provided as part of the lesson. Students may ask how the numbers in a random number table are generated. Today, computer programs are used to generate random number tables, but the idea behind these tables is relatively simple. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are placed in a bag or jar (with each digit written on a slip of paper or a chip). The 10 digits are thoroughly mixed. A slip or chip is picked. The number on the slip or chip is recorded and then returned to the bag or jar. This process is repeated, generating a random list of 0’s to 9’s.
Classwork Example 1 (5 minutes): Estimating a Population Mean Example 1: Estimating a Population Mean The owners of a gym have been keeping track of how long each person spends at the gym. Eight hundred of these times (in minutes) are shown in the population tables located at the end of the lesson. These 𝟖𝟖𝟖𝟖𝟖𝟖 times will form the population that you will investigate in this lesson.
Look at the values in the population. Can you find the longest time spent in the gym in the population? Can you find the shortest? On average, roughly how long do you think people spend at the gym? In other words, by just looking at the numbers in the two tables, make an estimate of the population mean.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
189 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
You could find the population mean by typing all 𝟖𝟖𝟖𝟖𝟖𝟖 numbers into a calculator or a computer, adding them up, and dividing by 𝟖𝟖𝟖𝟖𝟖𝟖. This would be extremely time-consuming, and usually it is not possible to measure every value in a population.
Instead of doing a calculation using every value in the population, we will use a random sample to find the mean of the sample. The sample mean will then be used as an estimate of the population mean.
In this example, students see a population of 800 times spent by people at a gym. The times are displayed in the population tables at the end of the lesson. Convey to students that it is not practical to use all 800 times to calculate the mean. It is important to make students aware of how unusual it would be to know all of the population values. Indeed, if you do have all the population values, then there is little point in taking a sample because you could calculate the population mean (using a computer if necessary). If there is time, ask the students to do the following:
Think of a population where you do not have all the population values.
Virtually any population will suffice in answer to this question; for example, the heights of all the adults in the town where the school is located.
Think of another population where you do have all the population values.
This is less common, but one example might be the salaries of all the employees of a company. All of the employee salaries will be stored in the company’s computer.
Discuss the questions in the example. Provide students time to think about and respond to the questions. The following responses provide a summary of the important points students should see as they work through the exercise.
Look at the values in the population. Can you find the longest time spent in the gym in the population? Can you find the shortest?
The longest time is 92 min., and the shortest time is 9 min. Students will provide other estimates, but direct them to these times.
On average, roughly how long do you think people spend at the gym? In other words, by just looking at the numbers in the two tables, make an estimate of the population mean.
It seems from the numbers in the tables that the mean time spent at the gym is around 60 min.
Example 2 (5 minutes): Selecting a Sample Using a Table of Random Digits Carefully discuss the sample directions outlined in this section. The goal is to recognize that each of the population values has an equal chance of being selected into the sample. Introduce the random number table, and as students work through this example, make sure that they understand how the table is used to identify individuals to be included in the sample. Once a sample is selected, the sample mean is calculated for this sample, and students are encouraged to think about how different samples would result in different values of the sample mean (this being an example of sampling variability).
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
190 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Example 2: Selecting a Sample Using a Table of Random Digits The table of random digits provided with this lesson will be used to select items from a population to produce a random sample from the population. The list of digits is determined by a computer program that simulates a random selection of the digits 𝟎𝟎, 𝟏𝟏, 𝟐𝟐, 𝟑𝟑, 𝟒𝟒, 𝟓𝟓, 𝟔𝟔, 𝟕𝟕, 𝟖𝟖, or 𝟗𝟗. Imagine that each of these digits is written on a slip of paper and placed in a bag. After thoroughly mixing the bag, one slip is drawn, and its digit is recorded in this list of random digits. The slip is then returned to the bag, and another slip is selected. The digit on this slip is recorded and then returned to the bag. The process is repeated over and over. The resulting list of digits is called a random number table.
Encourage students to think about the following question and to possibly share their suggestions prior to discussing the subsequent steps with all students. How could you use a table of random digits to take a random sample? Step 1: Place the table of random digits in front of you. Without looking at the page, place the eraser end of your pencil somewhere on the table. Start using the table of random digits at the number closest to where your eraser touched the paper. This digit and the following two specify which observation from the population tables will be the first observation in your sample. For example, suppose the eraser end of your pencil lands on the twelfth number in row 𝟑𝟑 of the random digit table. This number is 𝟓𝟓, and the two following numbers are 𝟏𝟏 and 𝟒𝟒. This means that the first observation in your sample is observation number 𝟓𝟓𝟓𝟓𝟓𝟓 from the population. Find observation number 𝟓𝟓𝟓𝟓𝟓𝟓 in the population table. Do this by going to Row 𝟓𝟓𝟓𝟓 and moving across to the column heading “𝟒𝟒.” This observation is 𝟓𝟓𝟓𝟓, so the first observation in your sample is 𝟓𝟓𝟓𝟓. If the number from the random number table is any number 𝟖𝟖𝟖𝟖𝟖𝟖 or greater, you will ignore this number and use the next three digits in the table.
Point out that four more selections are to be made to complete the random sample of five. Emphasize that the next four observations are determined by continuing on from the starting position in the random number table. The following directions and observations are important for students to understand: Step 2: Continue using the table of random digits from the point you reached, and select the other four observations in your sample like you did above. For example, continuing on from the position in the example given in Step 1:
The next number from the random digit table is 𝟕𝟕𝟕𝟕𝟕𝟕, and observation 𝟕𝟕𝟕𝟕𝟕𝟕 is 𝟔𝟔𝟔𝟔
The next number from the random digit table is 𝟓𝟓𝟓𝟓𝟓𝟓, and observation 𝟓𝟓𝟓𝟓𝟓𝟓 is 𝟑𝟑𝟑𝟑.
The next number from the random digit table is 𝟗𝟗𝟗𝟗𝟗𝟗, and there is no observation 𝟗𝟗𝟗𝟗𝟗𝟗. So, we ignore these three digits.
The next number from the random digit table is 𝟗𝟗𝟗𝟗𝟗𝟗, and there is no observation 𝟗𝟗𝟗𝟗𝟗𝟗. So, we ignore these three digits. The next number from the random digit table is 𝟑𝟑𝟑𝟑𝟑𝟑, and observation 𝟑𝟑𝟑𝟑𝟑𝟑 is 𝟕𝟕𝟕𝟕.
The next number from the random digit table is 𝟔𝟔𝟔𝟔𝟔𝟔, and observation 𝟔𝟔𝟔𝟔𝟔𝟔 is 𝟒𝟒𝟒𝟒.
Exercises 1–4 (8 minutes) Students should work independently on Exercises 1–4. Then, discuss and confirm as a class. The values found in Example 2 are used to illustrate the answers to the following questions, but students should select their own random sample of five.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
191 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Exercises 1–4 Initially, you will select just five values from the population to form your sample. This is a very small sample size, but it is a good place to start to understand the ideas of this lesson. 1.
Use the table of random numbers to select five values from the population of times. What are the five observations in your sample? Make sure students write down these values and understand how they were selected. In the example given, the following observations were selected: 𝟓𝟓𝟓𝟓 𝐦𝐦𝐦𝐦𝐦𝐦., 𝟔𝟔𝟔𝟔 𝐦𝐦𝐦𝐦𝐦𝐦., 𝟑𝟑𝟑𝟑 𝐦𝐦𝐦𝐦𝐦𝐦., 𝟕𝟕𝟕𝟕 𝐦𝐦𝐦𝐦𝐦𝐦., and 𝟒𝟒𝟒𝟒 𝐦𝐦𝐦𝐦𝐦𝐦.
2.
For the sample that you selected, calculate the sample mean. For the given example, the sample mean is
3.
𝟓𝟓𝟓𝟓+𝟔𝟔𝟔𝟔+𝟑𝟑𝟑𝟑+𝟕𝟕𝟕𝟕+𝟒𝟒𝟒𝟒 𝟓𝟓
= 𝟓𝟓𝟓𝟓. 𝟖𝟖 𝐦𝐦𝐦𝐦𝐦𝐦.
You selected a random sample and calculated the sample mean in order to estimate the population mean. Do you think that the mean of these five observations is exactly correct for the population mean? Could the population mean be greater than the number you calculated? Could the population mean be less than the number you calculated? Make sure that students see that the value of their sample mean (𝟓𝟓𝟓𝟓. 𝟖𝟖 minutes in the given example, but students will have different sample means) is not likely to be exactly correct for the population mean. The population mean could be greater or less than the value of the sample mean.
4.
In practice, you only take one sample in order to estimate a population characteristic. But, for the purposes of this lesson, suppose you were to take another random sample from the same population of times at the gym. Could the new sample mean be closer to the population mean than the mean of these five observations? Could it be farther from the population mean? Make sure students understand that if they were to take a new random sample, the new sample mean is unlikely to be equal to the value of the sample mean of their first sample. It could be closer to or further from the population mean.
Exercises 5–7 (7 minutes) Before students begin the following exercises, put all of their sample means on the board or poster paper. Then, ask the following questions: What do you notice about the sample means? Why do you think this is true? Anticipate that students will indicate that the sample means posted are different and that several of them are very close to each other. They may also point out how different some of them are from each other. Students should recognize that the samples come from the same population but that the actual selections can often be different. The different selections result in different sample means. Point out that the sample mean is an example of a sample statistic. The variation of sample statistics from sample to sample is called sampling variability.
Here, each student selects a random sample from the population and calculates the sample mean. Students should work individually, helping their neighbors as needed. The use of a calculator to calculate the sample mean should be encouraged to save time and increase accuracy. Point out that for this exercise, students should determine the mean to the nearest tenth of a minute.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
192 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Exercises 5–7 As a class, you will now investigate sampling variability by taking several samples from the same population. Each sample will have a different sample mean. This variation provides an example of sampling variability. 5.
Place the table of random digits in front of you, and without looking at the page, place the eraser end of your pencil somewhere on the table of random numbers. Start using the table of random digits at the number closest to where your eraser touches the paper. This digit and the following two specify which observation from the population tables will be the first observation in your sample. Write this three-digit number and the corresponding data value from the population in the space below. Answers will vary based on the numbers selected.
6.
Continue moving to the right in the table of random digits from the place you ended in Exercise 5. Use three digits at a time. Each set of three digits specifies which observation in the population is the next number in your sample. Continue until you have four more observations, and write these four values in the space below. Answers will vary based on the numbers selected.
7.
Calculate the mean of the five values that form your sample. Round your answer to the nearest tenth. Show your work and your sample mean in the space below. Answers will vary based on the numbers selected.
Exercises 8–11 (10 minutes) Exercises 8–11 You will now use the sample means from Exercise 7 from the entire class to make a dot plot. 8.
Write the sample means for everyone in the class in the space below. Answers will vary based on collected sample means.
9.
Use all the sample means to make a dot plot using the axis given below. (Remember, if you have repeated or close values, stack the dots one above the other.)
35
40
45
50 55 Sample Mean
60
65
70
10. What do you see in the dot plot that demonstrates sampling variability? Sample response: The dots are spread out indicating that the sample means are not the same. The results indicate what we discussed as sampling variability. (See explanation provided at the end of this lesson.)
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
193 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 17
7•5
11. Remember that in practice you only take one sample. (In this lesson, many samples were taken in order to demonstrate the concept of sampling variability.) Suppose that a statistician plans to take a random sample of size 𝟓𝟓 from the population of times spent at the gym and that he will use the sample mean as an estimate of the population mean. Approximately how far can the statistician expect the sample mean to be from the population mean? Answers will vary. Allow students to speculate as to what the value of the population mean might be and how far a sample mean would be from that value. The distance students indicate will be based on the dot plot. Students may indicate the sample mean to be exactly equal to the population and that the distance would be 𝟎𝟎. They may also indicate the sample mean to be one of the dots that is a minimum or a maximum of the distribution and suggest a distance from the minimum or maximum to the center of the distribution. Students should begin to see that the distribution has a center that they suspect is close to the population’s mean.
Here, the sample means for the class are combined in order to form a visual illustration of sampling variability (in the MP.2 form of a dot plot of the class’s sample means). Students are asked to show how the pattern in the dot plot supports the notion of sampling variability. Students will need to refer to the dot plot in their work in the next lesson, so every student will need to complete the dot plot. Exercise 10 summarizes an important part of this lesson. The fact that the sample mean varies from sample to sample is an example of sampling variability. Clearly emphasize this summary. Exercise 11 covers an important concept that is worth thinking about in advance of the class. When you take a sample and calculate the sample mean, you know that the value of the sample mean is very unlikely to be exactly equal to the population mean. But how close can you expect it to be? The dot plot constructed in this lesson provides a way to answer this question (at least for a random sample of size 5 from the population presented in this class). The population mean is approximately in the center of the dots in the dot plot constructed in Exercise 9. Indicate that the distances of the dots from the population mean provide information about how far the statistician can expect the sample mean to be from the population mean. An important point is also made at the beginning of Exercise 11, which is worth emphasizing whenever the opportunity arises in the remaining lessons. In practice, you only take one sample from a population. If you took two samples, then the sensible thing would be to combine them into one larger sample. We only take multiple samples in this lesson in order to demonstrate the concept of sampling variability.
Suppose that you are going to take one sample from a population. Once you have your sample, you will calculate the sample mean. Explain what is meant by sampling variability in this context.
Before taking your sample, you know that there are many different sets of individuals who could form your sample. Every different possible sample could give a different value for the sample mean. This is known as sampling variability.
It is important that students be aware that the idea of sampling variability does not apply only to the sample mean. Suppose, for example, an advertising agency plans to take a random sample of people and ask those people whether they have seen a particular commercial. The population is all the people who could be asked the question. The proportion of people in the sample who have seen the commercial is a sample statistic, and the value of this sample statistic will vary from sample to sample. This statistic also has sampling variability. Simulations will be developed in later lessons to illustrate sampling variability of proportions calculated using data from random samples of a population. Lesson 20 begins students’ investigations with the proportion as the statistic of interest.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
194 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Population
𝟏𝟏
𝟐𝟐
𝟑𝟑
𝟒𝟒
𝟓𝟓
𝟔𝟔
𝟕𝟕
𝟖𝟖
𝟗𝟗
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟎𝟎𝟎𝟎
𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒
𝟖𝟖𝟖𝟖
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟖𝟖𝟖𝟖
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 Lesson 17: Date:
𝟎𝟎
𝟎𝟎𝟎𝟎
𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐
𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟐𝟐𝟐𝟐
𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖
𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔 𝟐𝟐𝟐𝟐 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟗𝟗𝟗𝟗 𝟗𝟗
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟗𝟗𝟗𝟗 𝟗𝟗
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟏𝟏𝟏𝟏 𝟒𝟒𝟒𝟒
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
195 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Population, continued
𝟎𝟎
𝟐𝟐
𝟑𝟑
𝟒𝟒
𝟓𝟓
𝟔𝟔
𝟕𝟕
𝟖𝟖
𝟗𝟗
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑
𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓
𝟏𝟏𝟏𝟏
𝟓𝟓𝟓𝟓
𝟖𝟖𝟖𝟖
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
𝟔𝟔𝟔𝟔
𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒
𝟔𝟔𝟔𝟔
𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏
𝟔𝟔𝟔𝟔
𝟖𝟖𝟖𝟖
𝟕𝟕𝟕𝟕
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟕𝟕𝟕𝟕
𝟕𝟕𝟕𝟕
𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟒𝟒𝟒𝟒
𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑
𝟕𝟕𝟕𝟕
𝟔𝟔𝟔𝟔
𝟕𝟕𝟕𝟕
Lesson 17: Date:
𝟏𝟏
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑
𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟖𝟖𝟖𝟖 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑
𝟖𝟖𝟖𝟖 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓
𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔
𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟏𝟏𝟏𝟏 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟏𝟏𝟏𝟏 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟏𝟏𝟏𝟏 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟏𝟏𝟏𝟏 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟕𝟕𝟕𝟕 𝟏𝟏𝟏𝟏
𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟕𝟕𝟕𝟕 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓
𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔
𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟖𝟖𝟖𝟖 𝟓𝟓𝟓𝟓 𝟒𝟒𝟒𝟒 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓
𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟔𝟔𝟔𝟔 𝟓𝟓𝟓𝟓 𝟔𝟔𝟔𝟔 𝟒𝟒𝟒𝟒 𝟕𝟕𝟕𝟕 𝟔𝟔𝟔𝟔 𝟕𝟕𝟕𝟕 𝟓𝟓𝟓𝟓 𝟓𝟓𝟓𝟓 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 𝟓𝟓𝟓𝟓
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
196 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Table of Random Digits Row 𝟏𝟏
𝟔𝟔
𝟔𝟔
𝟕𝟕
𝟐𝟐
𝟖𝟖
𝟎𝟎
𝟎𝟎
𝟖𝟖
𝟒𝟒
𝟎𝟎
𝟎𝟎
𝟒𝟒
𝟔𝟔
𝟎𝟎
𝟑𝟑
𝟐𝟐
𝟐𝟐
𝟒𝟒
𝟔𝟔
𝟖𝟖
𝟒𝟒
𝟗𝟗
𝟏𝟏
𝟏𝟏
𝟗𝟗
𝟐𝟐
𝟖𝟖
𝟑𝟑
𝟎𝟎
𝟑𝟑
𝟔𝟔
𝟕𝟕
𝟕𝟕
𝟒𝟒
𝟕𝟕
𝟓𝟓
𝟗𝟗
𝟖𝟖
𝟏𝟏
𝟖𝟖
𝟑𝟑
𝟐𝟐 𝟑𝟑 𝟓𝟓 𝟔𝟔 𝟕𝟕 𝟖𝟖 𝟗𝟗
𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒
𝟖𝟖 𝟓𝟓 𝟗𝟗 𝟖𝟖 𝟒𝟒 𝟗𝟗 𝟑𝟑 𝟐𝟐 𝟎𝟎
𝟖𝟖 𝟒𝟒 𝟒𝟒 𝟔𝟔 𝟑𝟑 𝟔𝟔 𝟖𝟖 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟖𝟖 𝟒𝟒 𝟗𝟗 𝟑𝟑 𝟐𝟐 𝟎𝟎
𝟖𝟖 𝟒𝟒 𝟒𝟒 𝟔𝟔 𝟑𝟑 𝟔𝟔 𝟖𝟖 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟖𝟖 𝟒𝟒 𝟗𝟗
𝟎𝟎
𝟑𝟑 𝟎𝟎
𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟐𝟐 𝟔𝟔 𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟐𝟐 𝟔𝟔 𝟎𝟎
𝟑𝟑 𝟏𝟏 𝟎𝟎
𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟐𝟐 𝟔𝟔 𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟐𝟐 𝟔𝟔 𝟎𝟎
𝟑𝟑 𝟏𝟏 𝟎𝟎
𝟏𝟏 𝟏𝟏 𝟗𝟗
Lesson 17: Date:
𝟑𝟑 𝟓𝟓 𝟐𝟐 𝟒𝟒 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟓𝟓 𝟎𝟎
𝟔𝟔 𝟗𝟗 𝟖𝟖 𝟑𝟑 𝟕𝟕 𝟑𝟑 𝟓𝟓 𝟏𝟏 𝟐𝟐 𝟒𝟒 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟓𝟓 𝟎𝟎
𝟔𝟔 𝟗𝟗 𝟖𝟖 𝟑𝟑 𝟕𝟕 𝟑𝟑 𝟓𝟓 𝟏𝟏 𝟐𝟐 𝟒𝟒 𝟏𝟏 𝟐𝟐
𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟔𝟔 𝟗𝟗 𝟕𝟕 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟒𝟒 𝟖𝟖 𝟎𝟎
𝟖𝟖 𝟐𝟐 𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟗𝟗 𝟔𝟔 𝟗𝟗 𝟕𝟕 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟒𝟒 𝟖𝟖 𝟎𝟎
𝟖𝟖 𝟐𝟐 𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟗𝟗 𝟔𝟔 𝟗𝟗 𝟕𝟕
𝟏𝟏 𝟑𝟑 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟏𝟏 𝟗𝟗 𝟓𝟓 𝟎𝟎
𝟒𝟒 𝟓𝟓 𝟕𝟕 𝟎𝟎
𝟒𝟒 𝟖𝟖 𝟏𝟏 𝟑𝟑 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟏𝟏 𝟗𝟗 𝟓𝟓 𝟎𝟎
𝟒𝟒 𝟓𝟓 𝟕𝟕 𝟎𝟎
𝟒𝟒 𝟖𝟖 𝟏𝟏 𝟑𝟑 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟕𝟕 𝟏𝟏
𝟏𝟏 𝟔𝟔 𝟕𝟕 𝟔𝟔 𝟎𝟎
𝟑𝟑 𝟏𝟏 𝟗𝟗 𝟖𝟖 𝟖𝟖 𝟐𝟐 𝟑𝟑 𝟎𝟎
𝟖𝟖 𝟎𝟎
𝟏𝟏 𝟔𝟔 𝟖𝟖 𝟕𝟕 𝟔𝟔 𝟎𝟎
𝟑𝟑 𝟏𝟏 𝟗𝟗 𝟖𝟖 𝟖𝟖 𝟐𝟐 𝟑𝟑 𝟎𝟎
𝟖𝟖 𝟎𝟎
𝟏𝟏 𝟔𝟔 𝟖𝟖 𝟕𝟕 𝟔𝟔 𝟎𝟎
𝟑𝟑
𝟏𝟏 𝟑𝟑 𝟒𝟒 𝟖𝟖 𝟕𝟕 𝟐𝟐 𝟗𝟗 𝟗𝟗 𝟏𝟏 𝟖𝟖 𝟒𝟒 𝟒𝟒 𝟒𝟒 𝟖𝟖 𝟎𝟎
𝟏𝟏 𝟑𝟑 𝟑𝟑 𝟒𝟒 𝟖𝟖 𝟕𝟕 𝟐𝟐 𝟗𝟗 𝟗𝟗 𝟏𝟏 𝟖𝟖 𝟒𝟒 𝟒𝟒 𝟒𝟒 𝟖𝟖 𝟎𝟎
𝟏𝟏 𝟑𝟑 𝟑𝟑 𝟒𝟒 𝟖𝟖 𝟕𝟕 𝟐𝟐
𝟐𝟐 𝟏𝟏 𝟔𝟔 𝟐𝟐 𝟐𝟐 𝟗𝟗 𝟎𝟎 𝟑𝟑 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟐𝟐 𝟔𝟔 𝟖𝟖 𝟐𝟐 𝟏𝟏 𝟎𝟎 𝟔𝟔 𝟐𝟐 𝟐𝟐 𝟗𝟗 𝟎𝟎 𝟑𝟑 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟐𝟐 𝟔𝟔 𝟖𝟖 𝟐𝟐 𝟏𝟏 𝟎𝟎 𝟔𝟔 𝟐𝟐 𝟐𝟐 𝟗𝟗
𝟕𝟕 𝟕𝟕 𝟑𝟑 𝟖𝟖 𝟗𝟗
𝟎𝟎 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕 𝟔𝟔 𝟖𝟖 𝟑𝟑 𝟐𝟐 𝟒𝟒 𝟕𝟕 𝟕𝟕 𝟑𝟑 𝟑𝟑 𝟖𝟖 𝟗𝟗
𝟎𝟎 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕 𝟔𝟔 𝟖𝟖 𝟑𝟑 𝟐𝟐 𝟒𝟒 𝟕𝟕 𝟕𝟕 𝟑𝟑 𝟑𝟑 𝟖𝟖 𝟗𝟗
𝟎𝟎
𝟎𝟎 𝟐𝟐 𝟔𝟔 𝟗𝟗 𝟎𝟎 𝟑𝟑 𝟖𝟖 𝟗𝟗 𝟕𝟕 𝟕𝟕 𝟏𝟏 𝟐𝟐 𝟖𝟖 𝟗𝟗 𝟎𝟎 𝟎𝟎 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟗𝟗 𝟎𝟎 𝟑𝟑 𝟖𝟖 𝟗𝟗 𝟕𝟕 𝟕𝟕 𝟏𝟏 𝟐𝟐 𝟖𝟖 𝟗𝟗 𝟎𝟎 𝟎𝟎 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟗𝟗 𝟎𝟎 𝟑𝟑
𝟏𝟏 𝟓𝟓 𝟔𝟔 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟏𝟏 𝟑𝟑 𝟏𝟏
𝟎𝟎 𝟕𝟕 𝟗𝟗 𝟏𝟏 𝟏𝟏
𝟎𝟎 𝟏𝟏 𝟓𝟓 𝟕𝟕 𝟔𝟔 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟏𝟏 𝟑𝟑 𝟏𝟏
𝟎𝟎 𝟕𝟕 𝟗𝟗 𝟏𝟏 𝟏𝟏
𝟎𝟎 𝟏𝟏 𝟓𝟓 𝟕𝟕 𝟔𝟔 𝟓𝟓 𝟗𝟗 𝟗𝟗
𝟗𝟗 𝟓𝟓 𝟑𝟑 𝟓𝟓 𝟕𝟕 𝟎𝟎 𝟔𝟔 𝟗𝟗 𝟑𝟑 𝟏𝟏 𝟓𝟓 𝟏𝟏 𝟖𝟖 𝟎𝟎 𝟒𝟒 𝟗𝟗 𝟓𝟓 𝟕𝟕 𝟑𝟑 𝟓𝟓 𝟕𝟕 𝟎𝟎 𝟔𝟔 𝟗𝟗 𝟑𝟑 𝟏𝟏 𝟓𝟓 𝟏𝟏 𝟖𝟖 𝟎𝟎 𝟒𝟒 𝟗𝟗 𝟓𝟓 𝟕𝟕 𝟑𝟑 𝟓𝟓 𝟕𝟕 𝟎𝟎
𝟏𝟏 𝟏𝟏 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕 𝟐𝟐 𝟏𝟏
𝟎𝟎 𝟎𝟎 𝟔𝟔 𝟓𝟓 𝟒𝟒 𝟏𝟏 𝟔𝟔 𝟏𝟏 𝟏𝟏 𝟒𝟒 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕 𝟐𝟐 𝟏𝟏
𝟎𝟎 𝟎𝟎 𝟔𝟔 𝟓𝟓 𝟒𝟒 𝟏𝟏 𝟔𝟔 𝟏𝟏 𝟏𝟏 𝟒𝟒 𝟕𝟕 𝟐𝟐
𝟎𝟎 𝟕𝟕
𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟓𝟓 𝟗𝟗 𝟒𝟒 𝟏𝟏 𝟔𝟔 𝟑𝟑 𝟎𝟎 𝟗𝟗 𝟎𝟎 𝟐𝟐 𝟒𝟒 𝟕𝟕 𝟒𝟒 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟓𝟓 𝟗𝟗 𝟒𝟒 𝟏𝟏 𝟔𝟔 𝟑𝟑 𝟎𝟎 𝟗𝟗 𝟎𝟎 𝟐𝟐 𝟒𝟒 𝟕𝟕 𝟒𝟒 𝟗𝟗 𝟒𝟒 𝟓𝟓
𝟕𝟕 𝟕𝟕 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟔𝟔 𝟑𝟑
𝟎𝟎 𝟑𝟑 𝟕𝟕 𝟒𝟒 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟑𝟑 𝟕𝟕 𝟕𝟕 𝟓𝟓 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟔𝟔 𝟑𝟑
𝟎𝟎 𝟑𝟑 𝟕𝟕 𝟒𝟒 𝟓𝟓 𝟗𝟗 𝟗𝟗 𝟑𝟑 𝟕𝟕 𝟕𝟕 𝟓𝟓 𝟐𝟐 𝟔𝟔 𝟔𝟔 𝟔𝟔
𝟏𝟏 𝟏𝟏 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟕𝟕 𝟗𝟗 𝟓𝟓 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟗𝟗 𝟓𝟓 𝟑𝟑 𝟐𝟐 𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟕𝟕 𝟗𝟗 𝟓𝟓 𝟎𝟎 𝟏𝟏 𝟎𝟎 𝟗𝟗 𝟓𝟓 𝟑𝟑 𝟐𝟐 𝟏𝟏 𝟏𝟏 𝟗𝟗 𝟕𝟕 𝟐𝟐 𝟐𝟐 𝟕𝟕
𝟑𝟑 𝟔𝟔 𝟎𝟎 𝟓𝟓 𝟑𝟑 𝟏𝟏
𝟎𝟎 𝟓𝟓 𝟔𝟔 𝟑𝟑 𝟖𝟖 𝟖𝟖
𝟎𝟎 𝟎𝟎 𝟐𝟐 𝟑𝟑 𝟔𝟔 𝟖𝟖
𝟎𝟎 𝟓𝟓 𝟑𝟑 𝟏𝟏
𝟎𝟎 𝟓𝟓 𝟔𝟔 𝟑𝟑 𝟖𝟖 𝟖𝟖
𝟎𝟎 𝟎𝟎 𝟐𝟐 𝟑𝟑 𝟔𝟔 𝟖𝟖
𝟎𝟎 𝟓𝟓 𝟑𝟑 𝟏𝟏
𝟑𝟑 𝟓𝟓 𝟎𝟎 𝟑𝟑 𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟏𝟏 𝟒𝟒 𝟓𝟓 𝟒𝟒 𝟗𝟗 𝟗𝟗 𝟕𝟕 𝟒𝟒 𝟑𝟑 𝟓𝟓 𝟏𝟏 𝟎𝟎 𝟑𝟑 𝟏𝟏 𝟕𝟕 𝟗𝟗 𝟏𝟏 𝟒𝟒 𝟓𝟓 𝟒𝟒 𝟗𝟗 𝟗𝟗 𝟕𝟕 𝟒𝟒 𝟑𝟑 𝟓𝟓 𝟏𝟏 𝟎𝟎 𝟑𝟑 𝟏𝟏 𝟕𝟕
𝟓𝟓 𝟔𝟔 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟖𝟖 𝟏𝟏 𝟒𝟒 𝟒𝟒 𝟑𝟑 𝟏𝟏 𝟐𝟐 𝟎𝟎 𝟑𝟑 𝟔𝟔 𝟓𝟓 𝟔𝟔 𝟖𝟖 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟖𝟖 𝟏𝟏 𝟒𝟒 𝟒𝟒 𝟑𝟑 𝟏𝟏 𝟐𝟐 𝟎𝟎 𝟑𝟑 𝟔𝟔 𝟓𝟓 𝟔𝟔 𝟖𝟖 𝟏𝟏 𝟎𝟎 𝟎𝟎 𝟖𝟖
𝟑𝟑 𝟓𝟓 𝟗𝟗 𝟑𝟑 𝟗𝟗 𝟕𝟕
𝟎𝟎 𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟖𝟖 𝟑𝟑 𝟓𝟓 𝟑𝟑 𝟗𝟗 𝟑𝟑 𝟗𝟗 𝟕𝟕
𝟎𝟎 𝟐𝟐 𝟒𝟒 𝟒𝟒 𝟐𝟐 𝟗𝟗 𝟒𝟒 𝟓𝟓 𝟖𝟖 𝟑𝟑 𝟓𝟓 𝟑𝟑 𝟗𝟗 𝟑𝟑 𝟗𝟗 𝟕𝟕
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
197 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 17
7•5
Closing (4 minutes) If there is time, ask students these questions:
You have learned today about sampling variability. Do you want sampling variability to be large or small?
We want sampling variability to be small.
Why do you want sampling variability to be small?
Suppose you are using the sample mean to estimate a population mean. You want the sampling variability of the sample mean to be small, so that you can expect your value of the sample mean to be close to the population mean.
Lesson Summary A population characteristic is estimated by taking a random sample from the population and calculating the value of a statistic for the sample. For example, a population mean is estimated by selecting a random sample from the population and calculating the sample mean. The value of the sample statistic (e.g., the sample mean) will vary based on the random sample that is selected. This variation from sample to sample in the values of the sample statistic is called sampling variability.
Exit Ticket (6 minutes)
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
198 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ___________________________________________________
7•5
Date____________________
Lesson 17: Sampling Variability Exit Ticket Suppose that you want to estimate the mean time per evening students at your school spend doing homework. You will do this using a random sample of 30 students.
1.
Suppose that you have a list of all the students at your school. The students are numbered 1, 2, 3, …. One way to select the random sample of students is to use the random digit table from today’s class, taking three digits at a time. If you start at the third digit of row 9, what is the number of the first student you would include in your sample?
2.
Suppose that you have now selected your random sample and that you have asked the students how long they spend doing homework each evening. How will you use these results to estimate the mean time spent doing homework for all students?
3.
Explain what is meant by sampling variability in this context.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
199 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
7•5
Exit Ticket Sample Solutions Suppose that you want to estimate the mean time per evening students at your school spend doing homework. You will do this using a random sample of 𝟑𝟑𝟑𝟑 students.
1.
Suppose that you have a list of all the students at your school. The students are numbered 𝟏𝟏, 𝟐𝟐, 𝟑𝟑, …. One way to select the random sample of students is to use the random digit table from today’s class, taking three digits at a time. If you start at the third digit of row 𝟗𝟗, what is the number of the first student you would include in your sample? The first student in the sample would be student number 𝟐𝟐𝟐𝟐𝟐𝟐.
2.
Suppose that you have now selected your random sample and that you have asked the students how long they spend doing homework each evening. How will you use these results to estimate the mean time spent doing homework for all students? I would calculate the mean time spent doing homework for the students in my sample.
3.
Explain what is meant by sampling variability in this context. Different samples of students would result in different values of the sample mean. This is sampling variability of the sample mean.
Problem Set Sample Solutions 1.
Yousef intends to buy a car. He wishes to estimate the mean fuel efficiency (in miles per gallon) of all cars available at this time. Yousef selects a random sample of 𝟏𝟏𝟏𝟏 cars and looks up their fuel efficiencies on the Internet. The results are shown below.
a.
b.
𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟑𝟑𝟑𝟑 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐
Yousef will estimate the mean fuel efficiency of all cars by calculating the mean for his sample. Calculate the sample mean, and record your answer below. (Be sure to show your work.) 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟑𝟑𝟑𝟑 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 = 𝟐𝟐𝟐𝟐. 𝟗𝟗 𝟏𝟏𝟏𝟏
In practice, you only take one sample to estimate a population characteristic. However, if Yousef were to take another random sample of 𝟏𝟏𝟏𝟏 cars from the same population, would he likely get the same value for the sample mean? No, it is not likely that Yousef would get the same value for the sample mean.
c.
What if Yousef were to take many random samples of 𝟏𝟏𝟏𝟏 cars? Would all of the sample means be the same? No, he could get many different values of the sample mean.
d.
Using this example, explain what sampling variability means. The fact that the sample mean will vary from sample to sample is an example of sampling variability.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
200 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 17
NYS COMMON CORE MATHEMATICS CURRICULUM
2.
7•5
Think about the mean number of siblings (brothers and sisters) for all students at your school. a.
What do you think is the approximate value of the mean number of siblings for the population of all students at your school? Answers will vary.
b.
How could you find a better estimate of this population mean? I could take a random sample of students, ask the students in my sample how many siblings they have, and find the mean for my sample.
c.
Suppose that you have now selected a random sample of students from your school. You have asked all of the students in your sample how many siblings they have. How will you calculate the sample mean? I will add up all of the values in the sample and divide by the number of students in the sample.
d.
If you had taken a different sample, would the sample mean have taken the same value? No. A different sample would generally produce a different value of the sample mean. It is possible, but unlikely, that the sample mean for a different sample would have the same mean.
e.
There are many different samples of students that you could have selected. These samples produce many different possible sample means. What is the phrase used for this concept? Sampling variability.
f.
Does the phrase you gave in part (e) apply only to sample means? No. The concept of sampling variability applies to any sample statistic.
Lesson 17: Date:
Sampling Variability 2/6/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
201 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
